Axis Shift

Axis Shift:

Let us consider the simple third-order characteristic equation

As a first try, let s_n = s 2 and note that we obtain a Routh array without a zero occurring in the first column. However, upon setting the shifted variable sn equal to s 1, we obtain

Then the Routh array is established as

There are roots on the shifted imaginary axis that can be obtained from the auxiliary polynomial

The shifting of the v-plane axis to ascertain the relative stability of a system is a very useful approach, particularly for higher-order systems with several pairs of closed-loop complex conjugate roots.